Question: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{3}z)^{-3}}}{{(a^{3}z^{-3})^{-5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{3}z)^{-3} = (a^{3})^{-3}(z)^{-3}}$ On the left, we have ${a^{3}}$ to the exponent ${-3}$ . Now ${3 \times -3 = -9}$ , so ${(a^{3})^{-3} = a^{-9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{3}z)^{-3}}}{{(a^{3}z^{-3})^{-5}}} = \dfrac{{a^{-9}z^{-3}}}{{a^{-15}z^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-9}z^{-3}}}{{a^{-15}z^{15}}} = \dfrac{{a^{-9}}}{{a^{-15}}} \cdot \dfrac{{z^{-3}}}{{z^{15}}} = a^{{-9} - {(-15)}} \cdot z^{{-3} - {15}} = a^{6}z^{-18}$